Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Optimal
نویسنده
چکیده
Optimal preconditioners (those that provide a proper cluster at 1) are very important for the cg-like methods since they make them converge superlinearly. As is well known, for Toeplitz matrices generated by a continuous symbol, many circulant and circulant-like (related to different matrix algebras) preconditioners were proved to be optimal. In contrast, for multilevel Toeplitz matrices, there has been no proof of the optimality of any multilevel circulants. In this paper we show that such a proof is not possible since any multilevel circulant preconditioner is not optimal, in the general case of multilevel Toeplitz matrices. Moreover, for matrices not necessarily Toeplitz, we present some general results that enable us to prove that many popular structured preconditioners can not be optimal.
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تاریخ انتشار 2007